End (topology)

In general topology, an end of a topological space generalises the notion of "point at infinity" of the real line or plane.

An end of a topological space X is a function e which assigns to each compact set K in X some connected component with non-compact closure e(K) of the complement X – K in a compatible way, so that

If X is compact, then there are no ends.

[edit] Examples

• The real line R has two ends, which may be denoted ±∞. If K is a compact subset of R then by the Heine-Borel theorem K is closed and bounded. There are two unbounded components of R – K: if K is contained in the interval [a,b], they are the components containing (-∞,a) and (b,+∞). An end is a consistent choice of the left- or the right-hand component.
• The real plane R² has one end, ∞. If K is a compact, hence closed and bounded, subset of the plane, contained in a disk, then there is a single unbounded component of R² – K, the component containing the complement of the disc.

[edit] Compactification

Denote the set of ends of X by E(X) and let . We may topologise X ^* by taking as neighbourhoods of an end e the sets for compact K in X.

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